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Mirrors > Home > MPE Home > Th. List > oppgsubg | Structured version Visualization version GIF version |
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppggic.o | β’ π = (oppgβπΊ) |
Ref | Expression |
---|---|
oppgsubg | β’ (SubGrpβπΊ) = (SubGrpβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 19100 | . . 3 β’ (π₯ β (SubGrpβπΊ) β πΊ β Grp) | |
2 | subgrcl 19100 | . . . 4 β’ (π₯ β (SubGrpβπ) β π β Grp) | |
3 | oppggic.o | . . . . 5 β’ π = (oppgβπΊ) | |
4 | 3 | oppggrpb 19326 | . . . 4 β’ (πΊ β Grp β π β Grp) |
5 | 2, 4 | sylibr 233 | . . 3 β’ (π₯ β (SubGrpβπ) β πΊ β Grp) |
6 | 3 | oppgsubm 19330 | . . . . . . 7 β’ (SubMndβπΊ) = (SubMndβπ) |
7 | 6 | eleq2i 2821 | . . . . . 6 β’ (π₯ β (SubMndβπΊ) β π₯ β (SubMndβπ)) |
8 | 7 | a1i 11 | . . . . 5 β’ (πΊ β Grp β (π₯ β (SubMndβπΊ) β π₯ β (SubMndβπ))) |
9 | eqid 2728 | . . . . . . . . 9 β’ (invgβπΊ) = (invgβπΊ) | |
10 | 3, 9 | oppginv 19327 | . . . . . . . 8 β’ (πΊ β Grp β (invgβπΊ) = (invgβπ)) |
11 | 10 | fveq1d 6904 | . . . . . . 7 β’ (πΊ β Grp β ((invgβπΊ)βπ¦) = ((invgβπ)βπ¦)) |
12 | 11 | eleq1d 2814 | . . . . . 6 β’ (πΊ β Grp β (((invgβπΊ)βπ¦) β π₯ β ((invgβπ)βπ¦) β π₯)) |
13 | 12 | ralbidv 3175 | . . . . 5 β’ (πΊ β Grp β (βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯ β βπ¦ β π₯ ((invgβπ)βπ¦) β π₯)) |
14 | 8, 13 | anbi12d 630 | . . . 4 β’ (πΊ β Grp β ((π₯ β (SubMndβπΊ) β§ βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
15 | 9 | issubg3 19113 | . . . 4 β’ (πΊ β Grp β (π₯ β (SubGrpβπΊ) β (π₯ β (SubMndβπΊ) β§ βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯))) |
16 | eqid 2728 | . . . . . 6 β’ (invgβπ) = (invgβπ) | |
17 | 16 | issubg3 19113 | . . . . 5 β’ (π β Grp β (π₯ β (SubGrpβπ) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
18 | 4, 17 | sylbi 216 | . . . 4 β’ (πΊ β Grp β (π₯ β (SubGrpβπ) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
19 | 14, 15, 18 | 3bitr4d 310 | . . 3 β’ (πΊ β Grp β (π₯ β (SubGrpβπΊ) β π₯ β (SubGrpβπ))) |
20 | 1, 5, 19 | pm5.21nii 377 | . 2 β’ (π₯ β (SubGrpβπΊ) β π₯ β (SubGrpβπ)) |
21 | 20 | eqriv 2725 | 1 β’ (SubGrpβπΊ) = (SubGrpβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 βcfv 6553 SubMndcsubmnd 18748 Grpcgrp 18904 invgcminusg 18905 SubGrpcsubg 19089 oppgcoppg 19310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-subg 19092 df-oppg 19311 |
This theorem is referenced by: lsmmod2 19645 lsmdisj2r 19654 |
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